Optical communication systems are widely used for data communication. An optical coherent transportation link may transmit an optical signal, which may comprise two orthogonal polarization components (sometimes referred to as an X polarization and a Y polarization) with a same carrier frequency. The carrier frequency may refer to an optical wavelength supplied by a laser with phase noise. The propagation channel may comprise various components, such as one or more optical fibers (or fiber optics), amplifiers, and filters. Such components may be the sources of unwanted signal changes including, but not limited to, chromatic dispersion (CD), nonlinear phase noise, polarization mode dispersion (PMD), polarization dependent loss (PDL), polarization dependent gain, polarization rotation, and optical white Gaussian noise. For example, when propagating through the propagation channel, the optical signal may experience polarization state change, and its two polarization components may experience different losses due to PDL (e.g., PDL may make one polarization worse than the other). In this case, the overall performance of the optical system may be limited by the worst scenario. Thus, it is desirable to develop coding schemes that provide reliable optical communication despite the presence of unwanted signal changes.